How to make a kin selection model when marginal fitness is non-linear?

We observe that the Taylor-Frank method for making kin selection models when fitness w is a nonlinear function of a continuous actor’s phenotype y and the average phenotype z in its social environment requires w ( y, z ) to be differentiable (as a function of two variables, i.e., jointly in y and z ). This means that even if w ( y, z ) is non-linear globally, locally it must be close to linear, meaning that its graph must be well approximated by a plane. When more than two individuals interact, this assumption is only satisfied when the marginal fitness of the actor is a linear function of the fraction of individuals in its social environment that share its phenotype. This assumption sometimes fails for biologically important fitness functions, for instance in microbial data and the theory of repeated n-person games. In these cases, the Taylor-Frank methodology cannot be used, and a more general form of direct fitness must replace it, to decide when a social mutant allele can invade a monomorphic population.